Three essays in econometrics

In this dissertation, I would like to consider the efficient estimation of various models in the presence of heteroskedasticity of unknown form. The first essay focuses on mean sqaure errors comparison of linear regression model of hetetroskedasticity with unknown form. I compare higher order properties of the efficient estimators which include the GMM-type Cragg estimator, FGLS based on series and kernel estimations. The comparison is to calculate the approximate mean square errors of estimators using the Nagar type stochastic expansion. In the second essay, I consider the efficient estimation of partial linear regression model under heteroskedastictiy with unknown form. I propose an efficient estimator and prove it achieves Chamberlain’s (1992) semi-parametric efficiency bound. The new estimator I propose has the same first order asymptotic properties as Li’s (2000) estimator. My estimator has the potential advantage of analyzing the higher order asymptotics. The third essay considers the two-step series estimation method for generated regressors problem in context of semiparametric regression model under heteroskedastictiy of unknown form. I establish the root-n consistency and asymptotic normality results of the two-step series estimators. Compared to the double kernel estimator introduced by Stengos and Yan (2001), my estimator has some computational advantage and is more accurate in the sense of the asymptotic variance. Simulation results show that the two-step series estimator outperforms the double kernel estimator in terms of mean absolute bias and mean square error. The estimators considered in three essay involve the problem of choosing smoothing parameters. Therefore, I also demonstrate how to pick optimal smoothing parameters in each essay.